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arxiv: 2604.19227 · v1 · submitted 2026-04-21 · 💻 cs.SC · cs.MS· math.RA

Recognition: unknown

SignatureTensors.jl: A Package for Signature Tensors in Julia

Gabriel Riffo, Leonard Schmitz

Pith reviewed 2026-05-10 01:30 UTC · model grok-4.3

classification 💻 cs.SC cs.MSmath.RA
keywords signature tensorspathsJulia packageOSCARexact computationnumerical computationiterated integrals
0
0 comments X

The pith

SignatureTensors.jl is a Julia package that computes signature tensors of paths and supports both exact and numerical calculations through OSCAR integration.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents SignatureTensors.jl as a new tool in Julia for calculating signature tensors, which encode the iterated integrals along a path. It describes the package's core functions and provides examples to illustrate their application. The integration with the OSCAR system stands out because it permits exact computations using algebraic methods as well as standard numerical approximations. A reader might care if they work with path-based data and need reliable ways to extract geometric information from it without custom coding.

Core claim

The authors introduce the SignatureTensors.jl package for computing signature tensors of paths in Julia. They outline its main capabilities and demonstrate usage with examples. The package's compatibility with OSCAR allows it to perform both exact symbolic calculations and numerical computations on signatures.

What carries the argument

Signature tensor computation routines that calculate the truncated signature of a given path, leveraging OSCAR for exact arithmetic operations.

Load-bearing premise

The code in SignatureTensors.jl correctly implements the mathematical operations required to obtain signature tensors from input paths.

What would settle it

Compare the package output for the signature of a unit interval path against the analytically known values of its iterated integrals up to a given order.

read the original abstract

We introduce SignatureTensors.jl, a new package for computing signature tensors of paths in julia. We present its core functionality and demonstrate its use through illustrative examples. The package is compatible with the computer algebra system OSCAR, enabling both exact and numerical computations with signatures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces SignatureTensors.jl, a new Julia package for computing signature tensors of paths. It presents the core functionality, demonstrates usage through illustrative examples, and states compatibility with the OSCAR computer algebra system to support both exact and numerical computations.

Significance. If the implementation is correct, the package would supply a practical open-source tool for signature tensor computations within the Julia ecosystem, especially for OSCAR users requiring both symbolic and numerical modes. This could aid reproducibility in rough path theory and algebraic combinatorics, though the manuscript offers no new mathematical results and the significance depends on verification beyond the provided examples.

major comments (2)
  1. [Abstract] The abstract and core functionality description assert that the package correctly computes signature tensors and integrates with OSCAR for exact/numerical modes, but supply no implementation details, benchmarks, error analysis, or verification against analytic signatures or other libraries (e.g., esig).
  2. [Illustrative Examples] The illustrative examples demonstrate usage but provide no quantitative checks, such as matching known signature values for simple paths, numerical stability tests, or edge-case handling, leaving the central correctness claim unsubstantiated.
minor comments (2)
  1. Add a brief API reference or table of exported functions to improve usability for readers interested in adopting the package.
  2. Include references to foundational literature on signature tensors and related software to better contextualize the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on the manuscript. We address each major point below and indicate the revisions we will make to strengthen the presentation of the package's correctness.

read point-by-point responses

  1. Referee: [Abstract] The abstract and core functionality description assert that the package correctly computes signature tensors and integrates with OSCAR for exact/numerical modes, but supply no implementation details, benchmarks, error analysis, or verification against analytic signatures or other libraries (e.g., esig).

    Authors: The manuscript is a concise software announcement rather than a full technical report on algorithms. Core implementation details reside in the open-source package repository. We will revise the abstract and add a short dedicated subsection describing the high-level approach to signature tensor computation (leveraging iterated integrals and tensor algebra) along with explicit verification against analytically known cases, such as the signature of a linear path. Initial numerical comparisons and OSCAR integration tests will also be included; comprehensive cross-library benchmarks (e.g., versus esig) and full error analysis are noted as future work but exceed the scope of this introductory paper. revision: yes

  2. Referee: [Illustrative Examples] The illustrative examples demonstrate usage but provide no quantitative checks, such as matching known signature values for simple paths, numerical stability tests, or edge-case handling, leaving the central correctness claim unsubstantiated.

    Authors: We agree that the current examples focus on syntax and usage. In revision we will augment the examples section with quantitative checks: explicit computation of the signature for a straight-line path and direct comparison to the closed-form exponential expression; floating-point stability tests under varying precisions; and discussion of edge cases including the zero path and constant paths. These additions will directly substantiate the correctness claims within the manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: software package description with no derivations or fitted claims

full rationale

The manuscript is a straightforward introduction to the SignatureTensors.jl package and its OSCAR integration. It contains no mathematical derivations, no predictions of new quantities from fitted parameters, no uniqueness theorems, and no self-citation chains that bear load on any central claim. The content consists of core functionality descriptions and illustrative examples; correctness is asserted via implementation rather than derived from prior results within the paper itself. No step reduces to a self-definition or tautology by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper introduces no new free parameters, axioms, or invented entities; it is a description of a software package implementing existing mathematical concepts rather than a theoretical derivation.

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Reference graph

Works this paper leans on

80 extracted references · 17 canonical work pages

  1. [1]

    Pfeffer, Max and Seigal, Anna and Sturmfels, Bernd , journal=

  2. [2]

    2019 , publisher=

    Améndola, Carlos and Friz, Peter and Sturmfels, Bernd , journal=. 2019 , publisher=

    2019

  3. [3]

    2024 , publisher =

    The. 2024 , publisher =

    2024

  4. [4]

    2024 , eprint=

    An algebraic geometry of paths via the iterated-integrals signature , author=. 2024 , eprint=

    2024

  5. [5]

    2024 , KEYWORDS =

    Mignot, Raphael and Clausel, Marianne and Usevich, Konstantin , URL =. 2024 , KEYWORDS =

    2024

  6. [6]

    SIAM Journal on Applied Algebra and Geometry , volume =

    Clausel, Marianne and Diehl, Joscha and Mignot, Raphael and Schmitz, Leonard and Sugiura, Nozomi and Usevich, Konstantin , title =. SIAM Journal on Applied Algebra and Geometry , volume =. 2024 , doi =. https://doi.org/10.1137/23M159024X , abstract =

    work page doi:10.1137/23m159024x 2024

  7. [7]

    Reutenauer, C. , year=

  8. [8]

    Ast\'erisque, MR 83m , volume=

    Gromov's almost flat manifolds , author=. Ast\'erisque, MR 83m , volume=

  9. [9]

    and Lorenzi, M

    Pennec, X. and Lorenzi, M. , booktitle=. 2020 , publisher=

    2020

  10. [10]

    An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications , volume=

    Casas, Fernando and Murua, Ander , year=. An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications , volume=. Journal of Mathematical Physics , publisher=. doi:10.1063/1.3078418 , number=

    work page doi:10.1063/1.3078418

  11. [11]

    and Lyons, Terry and Seigal, Anna , title =

    Friz, Peter K. and Lyons, Terry and Seigal, Anna , title =. Bulletin of the London Mathematical Society , pages =. 2024 , doi =. https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/blms.13110 , abstract =

    work page doi:10.1112/blms.13110 2024

  12. [12]

    Decomposing tensor spaces via path signatures , journal =

    Carlos Améndola and Francesco Galuppi and Ángel David. Decomposing tensor spaces via path signatures , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.jpaa.2024.107807 , url =

    work page doi:10.1016/j.jpaa.2024.107807 2025

  13. [13]

    2024 , eprint=

    Rank and symmetries of signature tensors , author=. 2024 , eprint=

    2024

  14. [14]

    Bayesian Analysis , publisher =

    Patric Bonnier and Harald Oberhauser , title =. Bayesian Analysis , publisher =. 2024 , doi =

    2024

  15. [15]

    and Victoir, Nicolas B

    Friz, Peter K. and Victoir, Nicolas B. , year=. Multidimensional Stochastic Processes as Rough Paths: Theory and Applications , publisher=

  16. [16]

    Acta Applicandae Mathematicae , year =

    Joscha Diehl and Jeremy Reizenstein , title =. Acta Applicandae Mathematicae , year =. doi:10.1007/s10440-018-00227-z , url =

    work page doi:10.1007/s10440-018-00227-z

  17. [17]

    Proceedings of the London Mathematical Society , volume=

    Iterated integrals and exponential homomorphisms , author=. Proceedings of the London Mathematical Society , volume=. 1954 , publisher=

    1954

  18. [18]

    Annals of Mathematics , volume=

    Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula , author=. Annals of Mathematics , volume=. 1957 , publisher=

    1957

  19. [19]

    Algebraic Statistics, Vol

    Signature matrices of membranes , author=. Algebraic Statistics, Vol. 17 , year=

  20. [20]

    1990 , publisher=

    Representations of nilpotent Lie groups and their applications: Volume 1, Part 1, Basic theory and examples , author=. 1990 , publisher=

    1990

  21. [21]

    2025 , eprint=

    A Primer on the Signature Method in Machine Learning , author=. 2025 , eprint=

    2025

  22. [22]

    Principal geodesic analysis for time series encoded with signature features , author=

  23. [23]

    Horn and Vladimir V

    Roger A. Horn and Vladimir V. Sergeichuk , keywords =. Linear Algebra and its Applications , volume =. 2006 , issn =. doi:https://doi.org/10.1016/j.laa.2006.01.005 , url =

    work page doi:10.1016/j.laa.2006.01.005 2006

  24. [24]

    Technical University of Denmark , volume=

    The matrix cookbook , author=. Technical University of Denmark , volume=

  25. [25]

    Dokovic and Khakim D

    Dragomir Z. Dokovic and Khakim D. Ikramov , keywords =. On the congruence of square real matrices , journal =. 2002 , issn =. doi:https://doi.org/10.1016/S0024-3795(02)00300-2 , url =

    work page doi:10.1016/s0024-3795(02)00300-2 2002

  26. [27]

    Tensor-to-tensor models with fast iterated sum features , journal =

    Joscha Diehl and Rasheed Ibraheem and Leonard Schmitz and Yue Wu , keywords =. Tensor-to-tensor models with fast iterated sum features , journal =. 2026 , issn =. doi:https://doi.org/10.1016/j.neucom.2026.132884 , url =

    work page doi:10.1016/j.neucom.2026.132884 2026

  27. [28]

    2024 , publisher=

    Diehl, Joscha and Krieg, Richard , journal=. 2024 , publisher=

    2024

  28. [29]

    Meyer , journal =

    Carl D. Meyer , journal =. Generalized Inversion of Modified Matrices , urldate =

  29. [30]

    and Graham, Benjamin , title =

    Reizenstein, Jeremy F. and Graham, Benjamin , title =. ACM Trans. Math. Softw. , month = mar, articleno =. 2020 , issue_date =. doi:10.1145/3371237 , abstract =

    work page doi:10.1145/3371237 2020

  30. [31]

    2021 , note=

    Kidger, Patrick and Lyons, Terry , booktitle=. 2021 , note=

    2021

  31. [32]

    SIAM journal on matrix analysis and applications , volume=

    Means and averaging in the group of rotations , author=. SIAM journal on matrix analysis and applications , volume=. 2002 , publisher=

    2002

  32. [33]

    2012 , school=

    The expected signature of a stochastic process , author=. 2012 , school=

    2012

  33. [34]

    2021 , publisher=

    Salvi, Cristopher and Cass, Thomas and Foster, James and Lyons, Terry and Yang, Weixin , journal=. 2021 , publisher=

    2021

  34. [35]

    2023 , publisher=

    Cuchiero, Christa and Gazzani, Guido and Svaluto-Ferro, Sara , journal=. 2023 , publisher=

    2023

  35. [36]

    The Annals of Applied Probability , volume=

    Optimal stopping with signatures , author=. The Annals of Applied Probability , volume=. 2023 , publisher=

    2023

  36. [37]

    2020 , publisher=

    Guide to competitive programming , author=. 2020 , publisher=

    2020

  37. [38]

    2007 , publisher=

    Ebene Geometrie , author=. 2007 , publisher=

    2007

  38. [39]

    Mathematics of Control, Signals, and Systems , volume=

    Shortest-path recovery from signature with an optimal control approach , author=. Mathematics of Control, Signals, and Systems , volume=. 2025 , publisher=

    2025

  39. [40]

    preprint arXiv:1907.08423 , year=

    Insertion algorithm for inverting the signature of a path , author=. preprint arXiv:1907.08423 , year=

    work page arXiv 1907

  40. [42]

    International Conference on Geometric Science of Information , pages=

    Vaucher, R. International Conference on Geometric Science of Information , pages=. 2025 , organization=

    2025

  41. [43]

    Vaucher, R. D. 2025 , school=

    2025

  42. [44]

    Recent Advances in Econometrics and Statistics: Festschrift in Honour of Marc Hallin , pages=

    The insertion method to invert the signature of a path , author=. Recent Advances in Econometrics and Statistics: Festschrift in Honour of Marc Hallin , pages=. 2024 , publisher=

    2024

  43. [45]

    Linear algebra and its applications , volume=

    Canonical forms for complex matrix congruence and *congruence , author=. Linear algebra and its applications , volume=. 2006 , publisher=

    2006

  44. [46]

    SIAM review , volume=

    Tensor decompositions and applications , author=. SIAM review , volume=. 2009 , publisher=

    2009

  45. [47]

    SIAM Journal on Computing , volume=

    Dub. SIAM Journal on Computing , volume=. 1990 , publisher=

    1990

  46. [48]

    1997 , publisher=

    Ideals, varieties, and algorithms , author=. 1997 , publisher=

    1997

  47. [49]

    The Annals of Statistics , pages=

    Distinctness of the eigenvalues of a quadratic form in a multivariate sample , author=. The Annals of Statistics , pages=. 1973 , publisher=

    1973

  48. [50]

    Journal of Algebra , volume=

    Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms , author=. Journal of Algebra , volume=. 2008 , publisher=

    2008

  49. [51]

    Linear algebra and its applications , volume=

    A note on canonical forms for matrix congruence , author=. Linear algebra and its applications , volume=. 1996 , publisher=

    1996

  50. [52]

    Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation , pages=

    Msolve: A library for solving polynomial systems , author=. Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation , pages=

    2021

  51. [53]

    1999 , publisher=

    Faugere, Jean-Charles , journal=. 1999 , publisher=

    1999

  52. [54]

    Ph. D. Thesis, Math. Inst., University of Innsbruck , author=

  53. [55]

    Journal of the European Mathematical Society , volume=

    Inverting the signature of a path , author=. Journal of the European Mathematical Society , volume=

  54. [56]

    Journal of Functional Analysis , volume=

    Hyperbolic development and inversion of signature , author=. Journal of Functional Analysis , volume=. 2017 , publisher=

    2017

  55. [57]

    Advances in neural information processing systems , volume=

    Deep signature transforms , author=. Advances in neural information processing systems , volume=

  56. [60]

    IEEE transactions on pattern analysis and machine intelligence , volume=

    Persistence paths and signature features in topological data analysis , author=. IEEE transactions on pattern analysis and machine intelligence , volume=. 2018 , publisher=

    2018

  57. [61]

    Signature-Based Models in Finance

    Cuchiero, Christa and Gazzani, Guido and M \"o ller, Janka and Svaluto-Ferro, Sara. Signature-Based Models in Finance. Signature Methods in Finance: An Introduction with Computational Applications. 2026. doi:10.1007/978-3-031-97239-3_7

    work page doi:10.1007/978-3-031-97239-3_7 2026

  58. [62]

    A Primer on the Signature Method in Machine Learning

    Chevyrev, Ilya and Kormilitzin, Andrey. A Primer on the Signature Method in Machine Learning. Signature Methods in Finance: An Introduction with Computational Applications. 2026. doi:10.1007/978-3-031-97239-3_1

    work page doi:10.1007/978-3-031-97239-3_1 2026

  59. [63]

    Differential equations driven by rough paths: Ecole d'Et

    Lyons, Terry J and Caruana, Michael and L. Differential equations driven by rough paths: Ecole d'Et. 2007 , publisher=

    2007

  60. [64]

    2010 , publisher=

    Multidimensional stochastic processes as rough paths: theory and applications , author=. 2010 , publisher=

    2010

  61. [65]

    Journal of Algebraic Combinatorics , volume=

    Two-parameter sums signatures and corresponding quasisymmetric functions , author=. Journal of Algebraic Combinatorics , volume=. 2026 , publisher=

    2026

  62. [66]

    Advances in Applied Mathematics , volume=

    A topological approach to mapping space signatures , author=. Advances in Applied Mathematics , volume=. 2025 , publisher=

    2025

  63. [67]

    Harang and Samy Tindel , keywords =

    Joscha Diehl and Kurusch Ebrahimi-Fard and Fabian N. Harang and Samy Tindel , keywords =. On the signature of an image , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.spa.2025.104661 , url =

    work page doi:10.1016/j.spa.2025.104661 2025

  64. [68]

    Améndola, Carlos and Riffo, Gabriel and Schmitz, Leonard , journal=

  65. [69]

    Benchmarktools.jl , author=

  66. [70]

    Varieties of signature tensors

    Carlos Améndola, Peter Friz, and Bernd Sturmfels. Varieties of signature tensors . Forum of Mathematics, Sigma , 7, 2019

    2019

  67. [71]

    Signature varieties of splines

    Carlos Améndola, Felix Lotter, and Leonard Schmitz. Signature varieties of splines. preprint arXiv:2602.13011, to appear in Proceedings of the 2026 International Symposium on Symbolic and Algebraic Computation , 2026

    work page arXiv 2026

  68. [72]

    Learning barycenters from signature matrices

    Carlos Am \'e ndola and Leonard Schmitz. Learning barycenters from signature matrices. preprint arXiv:2509.07815 , 2025

    work page arXiv 2025

  69. [73]

    Computing Path Signature Varieties in Macaulay2

    Carlos Améndola, Angelo El Saliby, Felix Lotter, and Oriol Reig Fité. Computing Path Signature Varieties in Macaulay2 . preprint arXiv 2506.01429, to appear in Journal of Software for Algebra and Geometry , 2026

    work page arXiv 2026

  70. [74]

    The Barycenter in Free Nilpotent Lie Groups and Its Application to Iterated-Integrals Signatures

    Marianne Clausel, Joscha Diehl, Raphael Mignot, Leonard Schmitz, Nozomi Sugiura, and Konstantin Usevich. The Barycenter in Free Nilpotent Lie Groups and Its Application to Iterated-Integrals Signatures . SIAM Journal on Applied Algebra and Geometry , 8(3):519--552, 2024

    2024

  71. [75]

    The C omputer A lgebra S ystem OSCAR : A lgorithms and E xamples , volume 32 of Algorithms and C omputation in M athematics

    Wolfram Decker, Christian Eder, Claus Fieker, Max Horn, and Michael Joswig, editors. The C omputer A lgebra S ystem OSCAR : A lgorithms and E xamples , volume 32 of Algorithms and C omputation in M athematics . Springer, 1 edition, 8 2024

    2024

  72. [76]

    Tensor-to-tensor models with fast iterated sum features

    Joscha Diehl, Rasheed Ibraheem, Leonard Schmitz, and Yue Wu. Tensor-to-tensor models with fast iterated sum features. Neurocomputing , 675:132884, 2026

    2026

  73. [77]

    FRUITS: feature extraction using iterated sums for time series classification

    Joscha Diehl and Richard Krieg. FRUITS: feature extraction using iterated sums for time series classification . Data Mining and Knowledge Discovery , 38(6):4122--4156, 2024

    2024

  74. [78]

    Multidimensional stochastic processes as rough paths: theory and applications , volume 120

    Peter K Friz and Nicolas B Victoir. Multidimensional stochastic processes as rough paths: theory and applications , volume 120. Cambridge University Press, 2010

    2010

  75. [79]

    S ignatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU

    Patrick Kidger and Terry Lyons. S ignatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU . In International Conference on Learning Representations , 2021. https://github.com/patrick-kidger/signatory

    2021

  76. [80]

    Differential equations driven by rough paths: Ecole d'Et \'e de Probabilit \'e s de Saint-Flour XXXIV-2004

    Terry J Lyons, Michael Caruana, and Thierry L \'e vy. Differential equations driven by rough paths: Ecole d'Et \'e de Probabilit \'e s de Saint-Flour XXXIV-2004 . Springer, 2007

    2004

  77. [81]

    Learning Paths from Signature Tensors

    Max Pfeffer, Anna Seigal, and Bernd Sturmfels. Learning Paths from Signature Tensors . SIAM Journal on Matrix Analysis and Applications , 2019

    2019

  78. [82]

    Benchmarktools.jl, 2017

    Jarrett Revels et al. Benchmarktools.jl, 2017

    2017

  79. [83]

    Reizenstein and Benjamin Graham

    Jeremy F. Reizenstein and Benjamin Graham. Algorithm 1004: The Iisignature Library: Efficient Calculation of Iterated-Integral Signatures and Log Signatures . ACM Trans. Math. Softw. , 46(1), March 2020

    2020

  80. [84]

    An efficient algorithm for tensor learning

    Leonard Schmitz. An efficient algorithm for tensor learning. preprint arXiv:2512.14218 , 2025

    work page arXiv 2025